Yann Brenier
Abstrat. Theoneptofharmoni(orwave)mapsuptorearrangementis
introdued.Arelationisestablishedwiththesmoothsolutionsoftheisother-
malirrotationalinvisidgasdynamisequations.
Onintroduitlanotion d'appliationharmonique a rearrangementpres. On
etablit un lien ave les solutions regulieres des equations derivant la dy-
namiqueisothermedesgazsansvisositenitourbillon.
1. Review of somelassialonepts
1.1. Harmoni (or wave) maps. LetS >0, T >0, U =℄0;T[℄0;S[ and
D=T d
=(R=Z) d
theattorus. Amap(t;s)2U !X(t;s)2D isusuallyalled
aharmonimap(resp. awavemap)ifitisaritialpointofthefuntional
Z
U 1
2 (j
t X(t;s)j
2
+j
s X(t;s)j
2
)dtds;
(1.1)
where =1(resp. = 1), with respetto perturbationswithompat support
in U. (Herej:j denotestheEulidean normonR d
.) ThismeansthatX solvesthe
homogeneousLaplae(resp. wave)equation
tt X+
ss X =0:
(1.2)
Intheharmoniase=1,amapX isalledminimizingharmoniifitminimizes
(1.1)asitsvalueXj
U
alongtheboundaryisxed.
Remark. A natural lass for the \target" D of a harmoni (or wave) map is
thelass of generalRiemannian manifolds. TheaseD =T d
is fairlytrivial but
suÆient for our disussion. Compat Riemmanian manifolds without boundary
ould also be onsidered. However, ompat manifolds with boundaries or non
ompatmanifolds,inpartiulartheEulideanspaeR d
,wouldausediÆulties.
1991MathematisSubjetClassiation. Primary58E2058E3076N;Seondary35J35Q35.
Keywordsandphrases. Harmoniity,gasdynamis,rearrangement,vibratingstrings.
WorkpartlysupportedbytheaustrianprojetSTART(FWF-TEC-Y-137).
1.2. Laws,rearrangements, Moser'slemma. Let(A;da)beaprobability
spae (typially A = [0;1℄or A = T d
equipped with the Lebesgue measure da).
Forameasurablefuntion a2A!X(a)2D,wedenethe\law"ofX to be
(x)= Z
A
Æ(x X(a))da;
(1.3)
whihisaprobabilitymeasureonD, morepreiselydened by
Z
D
h(x)d(x)= Z
A
h(X(a))da;
(1.4)
forallh2C(D). (Anusualdenominationforis\push-forward"ofdabyX.) We
nowsaythattwosuhmeasurablefuntionsX andY areequaluptorearrangement
iftheyhavethesamelaw,namely
Z
A
Æ(x Y(a))da= Z
A
Æ(x X(a))da:
(1.5)
Letus quoteaveryusefulresult,due to MoserandlaterimprovedbyDaorogna
andMoser[DM℄,atleastinthespeialaseA=T d
Lemma 1.1. Let A =D =T d
. Let (x)> 0be a smooth funtion of x 2D
withmean1. ThenthereisasmoothorientationpreservingdieomorphismX from
A to D suhthat (x)dx isthe law of X. In addition, as depends smoothly on
someparameters2[0;1℄, wemaynd X thatalso depends smoothly ons.
Anelementaryproofisprovidedin setion8.8
1.3. Isothermal irrotational gas dynamis. The evolutionof aninvisid
gas moving in the Eulidean spae R d
(physially d = 1;2;3) is desribed by a
densityeld(s;x)>0,apressureeldp(s;x)>0,atemperatureeld(s;x)>0,
andaveloityeldu(t;x),valuedin R d
,wheresstandsforthetimevariableand
xforthespaevariable. Theseeldsaresubjettothefollowingequations(where
weusenotationsr for(
x
1
;::::;
x
d
)and: fortheEulideaninnerprodutinR d
)
s
+r:(u)=0;
(1.6)
whih expresses the onservation of mass, and is usually alled \the ontinuity
equation",
s
u+(u:r)u+ rp
=0;
(1.7)
whih is equivalent to the onservation of momentum. The evolution is alled
isothermalifthetemperatureeld isaonstant(s;x)=>0andthepressure
isgivenby
p=;
(1.8)
forsomeonstant>0. Introduing
= p
log; (1.9)
theseequationsanbewrittenasasymmetrirst-orderhyperbolisysteminspae
timevariables(x;s)
s
+u:r+ p
r:u=0;
(1.10)
s
u+(u:r)u+ p
r=0;
(1.11)
forwhihexisteneanduniquenessofsmoothsolutions,forsmoothinitialdataand
shorttime intervals,arestandardresults(see[Si℄forinstane). Potentialveloity
eldsofform
u=r;
(1.12)
with(salar)potentials(s;x)arepreservedduring theevolution,aslongasthey
staysmooth. Then,theontinuityequationbeomes
s
+r:(r)=0 (1.13)
andthemomentumequationanbeintegratedout,whihleads(uptoanirrelevant
onstant)to:
s +
1
2 jrj
2
+log=0:
(1.14)
Then,equations (1.13),(1.14), from aself-onsistentsystemofevolutionPDEsin
and,desribingisothermalirrotationalinvisidgasdynamis.
Remarkonthe Ationpriniple. Ataformallevel,equations(1.13),(1.14)an
beeasilyderivedbyvaryingtheAtion
Z
( jEj
2
2
log)dsdx;
(1.15)
withrespetto (s;x)>0,E(s;x)2R d
, subjetto
s
+r:E=0:
(1.16)
Indeed, (s;x) is introdued as a Lagrange multiplier for onstraint (1.16). By
varyingtheassoiatedLagrangian
Z
( jEj
2
2
log
s
r:E)dsdx;
with respet to (;E;), we get E = r and (1.14). Of ourse, as usual for
hyperboli equations,the Ationprinipledoesnotorrespond toaminimization
problem. Notie that, in the unphysial ase = 1, Ation (1.15) beomes a
onvexfuntional of (;E). Then, the orresponding hange of sign for log in
equation (1.14) transforms system (1.13), (1.14) into an ellipti system in spae
timevariables(x;s).
2. Harmoni orwave maps up to rearrangement
Letus nowombinethe oneptsofharmoni(or wave)maps andrearrange-
ments.
Definition2.1. Wesaythat
(t;x;a)2UA!X(t;s;a)2D;
isaharmoni map (resp. awave map) upto rearrangement, in shortaHMUR
(resp. aWMUR ),if itisaritial pointof
J(X)=inf
Y Z
UA 1
2 (j
t
X(t;s;a)j 2
+j
s
Y(t;s;a)j 2
)dtdsda;
(2.1)
withrespettoperturbations withompatsupportinU,wherethe inmumisper-
formedover all possible rearrangement Y of X with respet toa 2A, i.e. for all
funtionsY suhthat
Z
A
h(Y(t;s;a))da= Z
A
h(X(t;s;a))da;
(2.2)
holds true for all h2C(D), (t;s)2U. We all J(X)the Dirihlet integral of X
\up torearrangement"(even inthe ase= 1, forsimpliity).
Our main goal is to show that the equations governing the HUR maps are
related to Fluid Mehanis, and morespeially to the equations of isothermal
irrotationalinvisidgas. Inordertoestablishthislink,itisusefultointroduethe
\phase"densityf(t;s;x;)0assoiatedtoagivenmapX(t;s;a)anddenedby
f(t;s;x;)= Z
Æ(x X(t;s;a))Æ(
t
X(t;s;a))da;
(2.3)
for(t;s;x;)2UDR d
.
Definition2.2. We say that X(t;s;a) is a homogeneous map if the phase
densityf denedby(2.3) ist independentandfatorized
f(s;x;)=(s;x)G();
(2.4)
whereG>0isasmooth radiallysymmetri probability densityonR d
and>0a
probablitydensity on[0;S℄D.
Oneof ourmain result(Theorem5.1) will showapreise orrespondene be-
tweensmoothsolutionsofisothermal irrotationalinvisidgas dynamisequations
andsmoothHUR mapswhiharehomogeneousin thesenseof Denition2.2. In
addition, it will be shown that for mostof suh maps, the density G must be a
enteredGaussian lawwithuniformtemperature>0,namely
G()=(2) d=2
exp(
jj 2
2 ):
(2.5)
Amodel ofolletively vibrating strings. Theoriginalmotivationofthe
oneptofharmoniandwavemapsuptorearrangementwasthefollowing(arti-
ial,uptoourknowledge)mehanialmodelofvibratingstrings,whihorresponds
tothease = 1. Letusonsideraolletionofvibratingstringsparameterized
by
s2[0;S℄!X(t;s;a)
and labelled by a 2 A. The kineti energy of eah vibrating string is evaluated
individually
Z
S
0 1
2 j
t
X(t;s;a)j 2
ds
and thenintegrated in a2Aand t 2[0;T℄. This leadsto therstpartof J(X),
dened by(2.1). ThepotentialenergyofX isnotevaluatedindividuallyforeah
vibratingstringbytheusualformula
Z
S
1
2 j
s
X(t;s;a)j 2
ds;
butratherolletivelybyrearrangingthelabelsofthestringsforeahxedvalue
of(t;s),in ordertogetthelowestpossibleenergy. Inotherwords,weonsiderall
possibleolletions oftitiousvibratingstrings
s2[0;S℄!Y(t;s;a)
havingthesamespaerepartitionasX, i.e. satisfying(2.2),and omputein this
lasstheinmumof
Z
A Z
S
0 1
2 j
s
Y(t;s;a)j 2
dsda:
Integrating in t the resulting inmum leads to the seond part of J(X). As will
be shown, a WUR map is homogeneous, in the sense of denition 2.2, as the
strings vibrate with a olletive Gaussian distribution of veloities with uniform
temperature>0. A ruderversionofthis modelof vibratingstringshas been
onsideredin[Br2℄,wherethevariablesisdisretewithtwovalues,say0and1and
thepotentialenergyisdenedwithanitediereneinsteadofansderivative. The
orrespondingmodelislinkedtoinompressibleFluidMehanis(Eulerequations)
andPlasmaPhysis(Vlasov-Poissonequations).
3. Evaluationof the Dirihlet integral \up to rearrangement"
Our rst result gives amore tratable formula for the Dirihlet integral \up
torearrangement",involvingaseond-orderlinearelliptiequation,atleastinthe
asewhenthelawofX isasmoothfuntion >0.
Theorem 3.1. LetX(t;s;a)be amap UA!D withD=T d
. Assumeits
law
(t;s;x)= Z
Æ(x X(t;s;a))da (3.1)
tobeasmooth (stritly)positive funtion on UD. Then
J(X)= Z
UA 1
2 (j
t
X(t;s;a)j 2
+j
s
Y(t;s;a)j 2
)dtdsda (3.2)
whereY is denedby
s
Y(t;s;a)=(r)(t;s;Y(t;s;a)); Y(t;s=0;a)=X(t;0;a);
(3.3)
and for eah xed t and s, (t;s;x) is the unique solution, with zero mean, in
x2T d
,ofthe ellipti equation
r:(r)=
s : (3.4)
Aproofisprovidedinsetion8.1.
4. The HMUR and WMUR equations
OurseondresultprovidestheequationssatisedbysmoothHUR(orWUR )
mapsX withsmoothlaw>0.
Theorem 4.1. Let X(t;s;a) be a smooth map U A ! D with D = T d
.
Assumeitslaw
(t;s;x)= Z
Æ(x X(t;s;a))da (4.1)
to be a smooth (stritly) positive funtion on U D. Then X is HUR (resp.
WUR ) if and only if there are two smooth real funtions and q depending on
(t;s;x)2UD suhthat
s
+r:(r)=0;
s +
1
2 jrj
2
=q;
(4.2)
tt
X(t;s;a)=( rq)(t;s;X(t;s;a)); (t;s;x)= Z
Æ(x X(t;s;a))da;
(4.3)
with = 1 (resp. = 1). In addition, (4.3) an be expressed in terms of the
phasedensityf denedby (2.3)as:
t
f+:r
x
f rq:r
f =0;
Z
f(t;s;x;d)=(t;s;x):
(4.4)
The proof of Theorem 4.1 is provided in setion 8.2. Subsequently, we all
HMUR (resp. WMUR ) equations the ombination of (4.2) and either (4.3) or
(4.4),for =1(resp. = 1). Thestudy oftheinitial valueproblemis entirely
open. Theouplingbetweenequations(4.2)and(4.4)isverypeuliar-bothtand
splaytheroleofa\time"variable,tfor(4.4)andsfor(4.2)-and,ofourse,highly
nonlinear. However,inthespeialaseof homogeneoussolutions,in thesense of
Denition 2.2,theseequationsanberelatedtolassialFluidMehanis.
5. Linkbetween HMUR equations and gasdynamis
Theorem 5.1. Letusonsiderasmoothsolution(f;;)totheHMUR(resp.
WMUR ) equations (4.4), (4.2), where f isthe phasedensity dened by (2.3)for
somemap X.
Assumethat does not depend on t and that,for some s
0
2℄0;S[, thereis a non
degenerate maximum pointx
0
,i.e. ( 2
xixj (s
0
;x
0
))>0inthe sense of symmetri
matries.
Then X isahomogeneous mapin the senseof Denition2.2, if andonly if
f(s;x;)=(s;x)G();
where i)G isa enteredGaussian density(2.5) with uniform temperature >0,
ii) (;) does not depend on t and solve the isothermal irrotational invisid gas
dynamis equations (1.13),(1.14) with=1(resp. = 1).
Proof. Notierstthatthenondegenerayonditionisnotartiial. Indeed
(s;x) =1,(s;x) =0, f(s;x;)=G() isatrivialsolutionfor any hoie ofG,
evenwhenGisnotGaussian. Theproofisstraightforward. From(4.4),weget
G():r(s;x) rq(t;s;x)(s;x):rG()
andtherefore,sineandGare(stritly)positive,
:rlog(s;x) rq(t;s;x):rlogG()=0:
Sine G is assumed to be radiallysymmetri, rlogG() is proportionalto . It
followsrstthat
rlog(s;x)= rlogq(t;s;x);
forsomeonstant,whihannotbeequaltozero,beauseofthenondegeneray
ondition( 2
x
i x
j (s
0
;x
0
))>0whihimpliesthattherangeofr(s
0
;x)spansR d
.
Thus,wehave
q(t;s;x)= log(s;x);
(5.1)
upto anirrelevantadditiveonstant,for someonstant distintfrom 0. Next,
weget
( rlogG()):rlog(s;x)=0:
Sinetherangeofx!r(s
0
;x)spansR d
,weobtain
rlogG()=0;
anddeduethatGistheenteredGaussiandensity(2.5)withuniformtemperature
>0. Finally,using (4.2),weonludethat aneessaryandsuÆientondition
forasolutionX to theHMUR (resp. WMUR )equations tosatisfy(2.4)is that
doesnotdependontandsolvetheisothermalirrotationalinvisidgasdynamis
equations(1.13),(1.14)with=1(resp. = 1).
Remark. Somewhatsurprisingly,thephysialisothermalgasdynamisequa-
tions,with=1,arelinkedto HUR maps,nottoWURmaps. TheWUR maps
orrespond to isothermalgas dynamis equationswith = 1, in whih asethe
systemisunphysialandspae-timeellipti(withimaginaryspeedsofpropagation).
6. MinimizingHUR maps
In the harmoni ase = 1, we an provide a suffiient ondition for a
smooth map X to be a minimizing HUR map. In addition, we an relate the
orresponding minimization problem, whih is ertainly not onvex, to a linear
maximizationproblemwithonvexonstraintsthatanbeseenasitsdualproblem.
Theorem 6.1. Let (X;;;q) be a smooth solution to the HMUR equations
(4.2), (4.3), asinTheorem4.1, with=1andU =℄0;T[℄0;S[. Letusdenote by
X
jU
the restritionof X for (t;x)2U,a2A: Assume
4T 2
supf d
X
i=1;j=1
2
x
i x
j
q(t;s;x)
i
j
j (t;s;x)2UD; jj1g <
2
: (6.1)
Then,X isaminimizingHURmaponUA. Inaddition,theDirihletintegralof
X \uptorearrangement"J(X),denedby(2.1)isequaltothefollowingfuntional
J(X
jU )=sup
Z
UA
((t;s;X(t;s;a);a)n
t (t;s) (6.2)
+(t;s;X(t;s;a))n
s
(t;s))dH 1
(t;s)da
wheredH 1
isthe Hausdormeasurealong U,n=(n
t
;n
s
)(t;s)the externalnor-
mal for (t;s) 2U, and and are smooth funtions depending respetively on
(t;s;x;a)and(t;s;x) subjettothe pointwiseinequality
t +
1
2 jrj
2
+
s +
1
2 jrj
2
0:
(6.3)
Aproofwillbeprovidedin setion8.3.
7. Generationof minimizingHUR maps from isothermal gasdynamis
Finally,weanprovidearatherexpliitonstrutionofminimizingHURmaps,
whih are homogeneous in the sense of Denition 2.2, from smooth solutions to
the isothermal gas dynamis equations. Subsequently, = 1 is dropped in the
notations.
Theorem 7.1. Let (s;x) >0;(s;x) be a smooth solution of the isothermal
irrotational invisid gasdynamis equations (1.13), (1.14) for s 2[0;S℄ and x 2
D=T d
,withnormalizedtotalmass
Z
D
(s;x)dx=1:
(7.1)
For eah s, let y 2 D ! X
0
(s;y) 2 D a dieomorphism hosen (aording to
Moser'slemma1.1) withlaw (s;x)dx
Z
D
Æ(x X
0
(s;y))dy=(s;x):
(7.2)
Dene A = D [0;1℄
d+1
and solve, for eah xed s and a = (y;w) 2 A, the
autonomous seond-orderODE int2[0;T℄
tt
X(t;s;a)=( r
)(s;X(t;s;a));
(7.3)
withinitial onditions
X(t=0;s;a)=X
0
(s;y);
t
X(t=0;s;a)=V
0 (w);
(7.4)
wherea=(y;w) and
V
0
(w)=(os(2w
1
);::::;os(2w
d ))
p
2log(w
d+1 ) : (7.5)
Assume
4T 2
supf d
X
i=1;j=1
2
xixj
( log(s;x))
i
j
j0sS; x2D; jj1g <
2
: (7.6)
Then, X is a minimizing harmoni map up to rearrangement on U A, where
U =℄0;T[℄0;S[. In addition, X ishomogeneous inthe sense ofdenition 2.2.
Thisresultalsoestablishesanintriguingrelationshipbetween,ononeside,the
isothermal irrotationalinvisidgas dynamisequations, whih arehyperboliand
forwhih theationprinipledoesnotorrespond toaminimization problem (as
explainedinsetion1.3), and,ontheother side,theminimizationoftheDirihlet
integraluptorearrangementforpresribedDirihletboundarydataand thedual
problem(6.2)whih isalinearmaximizationproblemwithonvexonstraints.
Proof. Let us onsider the phase density f assoiated with X by (2.3). By
onstrutionof(X(0;s;a);
t
X(0;s;a)),through(7.2),(7.4),(7.5),f satises
f(t;s;x;)=(s;x)G(); G()=(2) d=2
exp(
jj 2
2 ) (7.7)
at time t=0. (Indeed, (s;x) is thelaw ofX
0
(s;y), by onstrutionof X
0 from
Moser'slemma,anditiswell-knownthattheGaussianlawanbewritten
G()= Z
[0;1℄
d+1
Æ( V
0
(w))dw;
(7.8)
whereV
0
isgivenby(7.5).) Beauseof(7.3),f satises
t
f+:r
x f+
r
:r
f =0:
(7.9)
where does not depend on t. Sine (s;x)G() is an obvious solution of this
equation, weonludethat (7.7) istrue forall t 2[0;T℄. Bysetting q= log,
weonludethat(X;;)solvetheHMUR equation(4.3),(4.2). FromTheorem
6.1,wededuethatX isaminimizingHURmapandfromTheorem5.1thatX is
homogeneousinthesenseof Denition2.2. Theproofof Theorem7.1istherefore
omplete.
8. Proofs
8.1. Proof ofTheorem3.1. Letusintrodue
j()=inf
Y Z
1
2 j
s
Y(t;s;a)j 2
dadtds;
(8.1)
whereY issubjetto
Z
Æ(x Y(t;s;a))da=(t;s;x);
(8.2)
andissupposedto besmoothand(stritly)positive. Letus prove
j()= Z
UD 1
2
(t;s;x)jr(t;s;x)j 2
dtdsdx;
(8.3)
where solves(3.4). Notiethat theminimization problem involvedin denition
(8.1)requiresnoboundaryonditions. Introdue
E(t;s;x)= Z
s
Y(t;s;a)Æ(x Y(t;s;a))da:
(8.4)
Observe that this vetor-valued measure E is absolutely ontinuous with respet
to . By Jensen's inequality, its (vetor-valued) density, denoted by e(t;s;x), is
squareintegrablewith
Z
1
2
je(t;s;x)j 2
(t;s;x)dtdsdx Z
1
2 j
s
Y(t;s;a)j 2
dadtds:
(8.5)
Inaddition,from(8.4),wededue
s
+r:(e)=0 (8.6)
(Indeed, forallt 2[0;T℄and allsmoothfuntions g(s;x) ompatlysupportedin
0<s<S :
Z
UD (
s
+e:r)dsdx= Z
UD (
s
+E:r)dsdx
= Z
(
s g+
s
Y(t;s;a):rg)(s;Y(t;s;a))dsda= Z
d
ds
(g(s;Y(t;s;a)))dsda=0:)
Thus,wegetalowerboundforj()byminimizingtheleft-handsideof(8.5)in e,
subjetto(8.6). Sine>0isxed,theuniquesolutionisgivenbye=r,where
(t;s;x) is, for eah xedt, s, theuniquesolution,with zeromean, in x2 T of
(3.4). Sowehave
j() Z
1
2 jrj
2
dtdsdx:
(8.7)
Next, letus dene Y(t;s;a)2 D =T d
for(t;s) 2U, a 2A, by(3.3) and hek
thatthelawofY,denoted by(t;~ s;x)anddened by
~
(t;s;x)= Z
Æ(x Y(t;s;a))da
doesnotdierfrom. First,~solvesthe\ontinuity"equation
s
~
+r:(~ r)=0:
(Indeed, forall testfuntion g(s;x), vanishing at s=0ands=S,for allxedt,
wehave
Z
(
s
g+r:rg)~dsdt= Z
(
s g+
s
Y(t;s;a):rg)(s;Y(t;s;a))dsda
= Z
d
ds
(g(s;Y(t;s;a)))dads=0:)
Ats=0,wehaveY(t;s=0;a)=X(t;0;a). Thus(t;~ s;x) =(t;s;x) at s=0,
sine is thelaw of X. By uniqueness of thesmooth solutionsto theontinuity
equation, we dedue that ~= for all s 2 [0;S℄. So, X and Y are equal up to
rearrangementwithlaw. Itfollowsthat
j() Z
1
2 j
s
Y(t;s;a)j 2
dadtds
(bydenition(8.1))
= Z
1
2
jr(t;s;Y(t;s;a)j 2
dadtds
(beauseof(3.3))
= Z
1
2
jr(t;s;x)j 2
(t;s;x)dtdsdx
(sineisthelawofY)
j()
(by(8.7)),whihompletestheproof.
8.2. Proof ofTheorem4.1. . FromTheorem3.1,wealreadyknowthatan
optimalY assoiated withX anbedened by (3.3),where r solvesthelinear
elliptiequation(3.4),withdatadependingon. Next,goingbaktotheoriginal
denition ofHUR(or WUR )maps, aneessaryand suÆientonditionforX to
beaHUR (oraWUR ) mapisthat(X;Y;q)isasaddlepointfortheLagrangian
Z
1
2 (j
t Xj
2
+j
s Yj
2
)dtdsda Z
(q(t;s;X(t;s;a)) q(t;s;Y(t;s;a)))dtdsda;
(8.8)
whereq(t;s;x)istheLagrangemultiplierfor onstraint(2.2). Weget
tt
X(t;s;a)=( rq)(t;s;X(t;s;a));
(8.9)
ss
Y(t;s;a)=(+rq)(t;s;Y(t;s;a)):
(8.10)
Therefore,r,denoted bye,mustsatisfy
s
e+(e:r)e=rq:
(8.11)
(Indeed, we have, for all t 2 [0;T℄ and all test funtion g(s;x) vanishing about
s=0ands=T,
Z
rqgdsdt= Z
ss
Y(t;s;a)g(s;Y(t;s;a))dsda
(using(8.10)andthefatthat isthelawofY)
= Z
s
Y(t;s;a) d
ds
(g(s;Y(t;s;a)))dsda= Z
(
s
g+r:rg)rdsdx
(beauseof(3.3),andusingagainthatisthelawofY). Thus,e=r satises
s
(e)+r:(ee)=rq
and, therefore, (8.11), sine>0satises (1.13).) Next,byintegration of (8.10)
inx,weget
s +
1
2 jrj
2
=q;
(8.12)
uptotheadditiontoq(t;s;x)ofanirrelevantfuntionof(t;s). So,wehaveobtained
thedesiredequations(4.2),(4.3)forX;;;q. Finally,equation(4.4)forthephase
densityffollowsfrom(4.3)byastandardargument. (Indeed,forallxeds2[0;S℄
andalltestfuntiong(t;x;)vanishingat t=0andt=T,wehave
Z
(
t
g+:r
x
g rq:r
g)fdtdxd
= Z
((
t
g)(t;X(t;s;a);
t
X(t;s;a))+
t
X(t;s;a):(rg)(t;X(t;s;a);
t
X(t;s;a))
(rq)(t;X(t;s;a)):(r
g)(t;X(t;s;a);
t
X(t;s;a)))dtda
= Z
d
dt
(g(t;X(t;s;a);
t
X(t;s;a)))dtda=0;
sine
tt
X(t;s;a) = (rq)(t;s;X(t;s;a)).) The proof of Theorem 4.1 is now
omplete.
8.3. Proof ofTheorem6.1. Weusetwointermediaryresults:
proposition8.1. LetX(t;s;a)beasmoothfuntiononUAvaluedinD=
T d
. Let J(X)be denedby (2.1) as the Dirihlet integral \up to rearrangement"
ofX. ThenJ(X)J(X
jU
)whereJ(X
jU
)isdenedby(6.2)asinTheorem 6.1.
proposition8.2. LetX(t;s;a)asmoothfuntiononUAvaluedinD=T d
,
and satisfying the assumptions of Theorem 6.1, inluding ondition (6.1) on T.
Then
J(X
jU )
1
2 Z
UA (j
t
X(t;s;a)j 2
+j
s
Y(t;s;a)j 2
)dtdsda;
(8.13)
whereJ(X )isdenedby (6.2)andY issomerearrangement ofX ina2A.
By ombining Propositions 8.1 and 8.2, we see that, under the assumptions
of Proposition 8.2, X is a minimizing harmoni map up to rearrangement and,
therefore,Theorem6.1follows. Inaddition,weget
J(X)=J(X
jU );
(8.14)
whih relates X to the linear maximization problem (6.2) in (;) with onvex
onstraint(6.3).
8.4. ProofofProposition8.1. LetX(t;s;a)andY(t;s;a)tobeadmissible
fordenition(2.1),whihmeansthatX andY havethesamelaw
(t;s;x)= Z
Æ(x X(t;s;a))da= Z
Æ(x Y(t;s;a))da (8.15)
and
Z
(j
t
X(t;s;a))j 2
+j
s
Y(t;s;a))j)dtdsda<+1:
(8.16)
Letusintrodue
(t;s;x;a)=Æ(x X(t;s;a));
(8.17)
Q(t;s;x;a)=
t
X(t;s;a)Æ(x X(t;s;a)) (8.18)
E(t;s;x)= Z
s
Y(t;s;a)Æ(x Y(t;s;a))da:
(8.19)
As in the proof of Theorem3.1, in setion 8.1, E is absolutely ontinuous with
respettoanditsdensity, e(t;s;x)satises
Z
1
2
je(t;s;x)j 2
d(t;s;x) Z
1
2 j
s
Y(t;s;a)j 2
dadtds:
(8.20)
The left-hand side an be written as a onvex funtion of (;e), beause of the
following(elementary)lemma(see[Br1℄):
Lemma 8.1. Let (;m) apair of measureson someompat spae M,respe-
tively valuedin RandR d
. Dene
K(;m)=supf Z
(y)d(y)+(y):dm(y); (y)+ 1
2 j(y)j
2
0; 8y2Mg;
(8.21)
wherethesupremumisperformedoverallontinuousfuntionsonM ,,respe-
tivelyvaluedinRandR d
. ThenK=+1unless isnonnegative, misabsolutely
ontinuouswithrespetto withavetor-valueddensityvwhihissquareintegrable
withrespetto. Inaddition, asK<+1,
K(;m)= 1
2 Z
jv(y)j 2
d(y):
(8.22)
Thusweanwrite(8.20) as:
K(;E) Z
1
2 j
s
Y(t;s;a)j 2
dtdsda:
(8.23)
Moresimply,Qisalsoabsolutelyontinuouswithrespetto and
K(;Q) Z
1
2 j
t
X(t;s;a)j 2
dtdsda:
(8.24)
(Asamatteroffat,wehaveanequality,butweneednotit.) Letusnowobserve
thatthepairs(;Q)and(;E)satisfythefollowingintegralidentitiesforallsmooth
funtions (t;s;x;a)and(t;s;x):
Z
UDA (
t
d+r:dQ)= (8.25)
Z
UA
(t;s;X(t;s;a);a)n
t
(t;s)dH 1
(t;s)da;
Z
UD (
s
d+r:dE)= Z
UA
(t;s;X(t;s;a))n
s
(t;s)dH 1
(t;s)da (8.26)
(whereweusethat doesnotdepend onaand Y hasthe samelawasX). Also
notiethat(8.15) anbeexpressedby
Z
UD
q(t;s;x)d(t;s;x)= Z
UA
q(t;s;x)d(t;s;x;a) (8.27)
foralltestfuntionq. Combiningalltheseproperties,wenallydeduethat
J(X)J(Xj
U );
wherewedene
J(Xj
U
)=infK(;Q)+K(;E);
with(;Q)and(;E)subjettoonstraints(8.25),(8.26)and(8.27). Notiethat
theseonstraintsonlyinvolvethevaluesofX(t;s;a)for(t;x)2U whihjusties
ournotationforJ. Equivalently,weanwrite
J(Xj
U
)= inf
;Q;;E sup
;;q;;;~;
~
Z
[ (t;~ s;x)
s
(t;s;x)+q(t;s;x))d(t;s;x) (8.28)
+(
~
(t;s;x) r(t;s;x)):dE(t;s;x)℄+ Z
[((t;s;x;a)
t
(t;s;x;a)
q(t;s;x))d(t;s;x;a)+((t;s;x;a) r(t;s;x;a)):dQ(t;s;x;a)℄
Z
UA
[(t;s;X(t;s;a);a)n
t
(t;s)+(t;s;X(t;s;a))n
s
(t;s)℄dH 1
(t;s)da;
where,,,~
~
aresubjetto
(t;s;x;a)+ 1
2
j(t;s;x;a)j 2
0; (t;~ s;x)+ 1
2 j
~
(t;s;x)j 2
0:
(8.29)
Thisisthesaddle-pointformulationofaonvexminimizationproblemwithonvex
onstraints.
ProvingProposition8.1isnowveryeasy. Letusset
=
t
+q; =r; ~=
s
q;
~
=r
and dene q by (8.12) where (;) are assumedto satisfy(6.3). This provides a
boundfrombelowforJ(Xj
U
). ThisboundisexatlyJ(Xj
U
),asdenedby(6.2).
ThustheproofofProposition8.1isomplete.
Remark. Byusing Rokafellar'sdualitytheorem(asstatedin [Brz ℄,hapter
1,forinstane),weouldshowthat J(Xj
U
)=J(Xj
U
),whihmeansthat there
isno\dualitygap".
8.5. Proof of Proposition 8.2. Let(;;X)to satisfy theassumptions of
Theorem 6.1. Let us rst introdue Y(t;s;a) 2 D = T d
dened or (t;s) 2 U,
a2A, by
s
Y(t;s;a)=(r)(t;s;Y(t;s;a)); Y(t;s=0;a)=X(t;0;a):
(8.30)
AsintheproofofTheorem3.1, thelawofY is . Next,introdue
(t;s;x;a)= inf
fz(t)=xg Z
t
0 (
1
2 jz
0
()j 2
q(;s;z()))d +z(0):
t
X(0;s;a);
(8.31)
where theinmum isperformedoverall smoothpaths 2[0;t℄!z()2D suh
thatz(t)=x. Letusstatetwolemmata
Lemma 8.2. Thefollowing identityistrue(undertheassumptionsofTheorem
6.1)
Z
UA
(t;s;X(t;s;a))n
s
(t;s))dH 1
(t;s)da (8.32)
= Z
1
2 j
s Yj
2
dtdsda+ Z
qdxdtds;
whereq isdenedby (8.12).
Lemma 8.3. Underondition(6.1),for allt2[0;T℄,theinmumindenition
(8.31)isahievedbyauniquepathz(),0 t,uniquesolutionofthetwopoint
mixedboundaryvalue problem (halfDirihlet, half Neumann)
z 00
()=( rq)(;s;z()); z(t)=x; z 0
(0)=
t
X(0;s;a):
(8.33)
LetuspostponetheproofofLemma8.2,Lemma8.3,andontinuetheproofof
Proposition 8.2. Itfollowsfrom lassialtheoryofHamilton-Jaobiequations(see
[Ba℄forinstane)that,underondition(6.1),(t;s;x;a)isasmoothsolution,for
0tT,totheHJequation
t +
1
2 jrj
2
+q=0:
(8.34)
Thus,bydenition(8.12)ofq, andsatisfyonstraint(8.29). Inaddition,using
(8.33) in the speialase x = X(t;s;a), wesee that z() =X(;s;a) beause of
(8.9)and,therefore,
(t;s;X(t;s;a);a)= Z
t
0 (
1
2 j
t
X(;s;a)j 2
q(;s;X(;s;a)))d + (8.35)
+X(0;s;a):
t
X(0;s;a):
Dierentiationg(8.35)withrespetto tgives
t
((t;s;X(t;s;a);a))= 1
2 j
t
X(t;s;a)j 2
q(t;s;X(t;s;a)):
Integratingthisexpressionwithrespetto(t;s;a)overUAleadsto
Z
(t;s;X(t;s;a);a)n
t
(t;s)dH 1
(t;s)da= Z
( 1
2 j
t Xj
2
q(t;s;X))dtdsda:
= 1
2 j
t Xj
2
dtdsda qdxdtds
(sineisthelawofX). Byaddingthisequalitytoidentity(8.32),wehavenally
obtained
Z
UA
((t;s;X(t;s;a);a)n
t
(t;s)+(t;s;X(t;s;a))n
s
(t;s))dH 1
(t;s)da
= 1
2 Z
(j
t Xj
2
+j
s Yj
2
)dtdsda:
Fromdenition (6.2), we dedue(8.13) and the proof of Proposition 8.2 is om-
pleted.
8.6. Proof ofLemma8.2. Wehave
Z
UA
(t;s;X(t;s;a))n
s
(t;s)dH 1
(t;s)da=
Z
UA
(t;s;Y(t;s;a))n
s
(t;s)dH 1
(t;s)da
(beauseX andY areequaluptorearrangement)
= Z
UA
s
((t;s;Y(t;s;a)))dtdsda
(byintegrationin s)
= Z
UA (
s
(t;s;Y(t;s;a))+
s
Y(t;s;a):r(t;s;Y(t;s;a)))dtdsda
= Z
UA (
s
(t;s;Y(t;s;a))+jr(t;s;Y(t;s;a))j 2
)dtdsda
(beauseof8.30)
= Z
(
s
+jrj 2
)dxdtds
(sineisthelawofY)
= Z
( 1
2 jrj
2
+q)dxdtds
(bydenition(8.12). Wealsohave
Z
1
2 jrj
2
dxdtds= Z
jr(t;s;Y(t;s;a)j 2
dtdsda= Z
j
s
Y(t;s;a)j 2
dtdsda
(using (8.30)). Thus, identity (8.32) follows and the proof of Lemma 8.2 is now
omplete.
8.7. Proof of Lemma 8.3. Let us x (t;s) 2 [0;T℄[0;S℄, x 2 D = T ,
a2A, anddene
h(z)= Z
t
0 (
1
2 jz
0
()j 2
q(;s;z()))d +z(0):
t
X(0;s;a);
(8.36)
forall path 2[0;t℄!z()2D suh that z(t)=x. Theseond derivativeofh
withrespetto zisgivenby
D 2
h(z)(~z;z)~ = Z
t
0 (j~z
0
()j 2
d
X
i;j=1 (
2
x
i x
j
q)(;s;z())~z
i ()~z
j ())d
This quadratiform is positivedenite, under ondition(6.1), beauseof thefol-
lowing(modied) Poinareinequality
2
Z
t
0 j~z()j
2
d 4t 2
Z
t
0 j~z
0
()j 2
d;
(8.37)
whihholdstrueforallt>0andforallsmoothfuntionz~suhthatz~ 0
(0)=0and
~
z(t)=0. (ThisPoinareinequality anbeestablishedbyusing Fourierseries,the
inequalitybeingsaturatedby ~z()=sin(
2t
).) It follows that ondition(6.1)on
T issuÆienttoenfore that,aslongastT,theinmumin denition (8.31)is
ahievedbyauniquepathz(),0 t,uniquesolutionofthetwopointmixed
boundaryvalueproblem (halfDirihlet,half Neumann)(8.33). Thus, theproofof
Lemma8.3isomplete.
8.8. Appendix : a proof of Moser's lemma. In the ase of the torus,
the proof is very easy. Firstsolve the Laplae equation (withperiodi boundary
onditions)
= 1
onthetorus(byusing Fourierseries,forinstane). Nextdene
v(;x)= r(x)
~ (;x)
; (;~ x)=(1 )+(x);
for 2[0;1℄andx2D,sothat
~
+r:(v)~ =0:
Then,foreaha2D solvetheinitialvalueproblem
Z(;a)=v(;Z(;a)); Z(0;a)=a; 8a2A=D:
Chek that,for eah 2 [0;1℄, Z(t;:) is asmooth dieomorphism ofD with law
~
(;x)dx. ConludebysettingX(a)=Z(1;a).
Aknowledgements. The author thanks TICAM, Austin, and the Erwin
Shrodinger Institut, Vienna, for their hospitality during the preparation of this
work. HealsothanksMihelRaslefor stimulatingdisussionsaboutationprin-
iplesinFluidMehanis.
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Math.52(1999)411-452.
[Br2℄ Y.Brenier, Derivation of the Eulerequations froma ariature of Coulombinteration,
Comm.Math.Phys.212(2000)93-104.
[Brz℄ H.Brezis,Analysefontionnelleappliquee,Masson,Paris,1974.
[DM℄ B.Daorogna,J.Moser,OnapartialdierentialequationinvolvingtheJaobiandetermi-
nant,Ann.Inst.H.PoinarAnal.NonLinaire7(1990)1-26.
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equationsandtheinompressiblelimit,IndianaUniv.Math.J.40(1991)535-550.
CNRS,Laboratoire
Dieudonn
e,
Universit
edeNie,Frane(onleavefromUniver-
sit
eParis6)
E-mailaddress: breniermath.unie.fr
